For a square plot with a 3m wide path running along all sides inside, the inner side = outer side − 2×3 = outer side − 6m. The area of the path = Outer area − Inner area. For example, if the plot is 30m × 30m, inner area = 24m × 24m, and path area = 900 − 576 = 324 m².
For a square plot side a with 3m path: inner side = a − 6
Area of path = Outer area − Inner area
Area of path = 12(a − 3) for 3m wide path in square
For 30m plot: path area = 12 × 27 = 324 m²
For 20m plot: path area = 12 × 17 = 204 m²
Path width is subtracted from both sides, so total subtraction = 2 × width
For a general path width w: inner side = a − 2w
Area of path = a² − (a−2w)² = 4w(a−w)
Given: A square plot with side = a metres. A path 3m wide runs along all sides on the inside.
Visualization:
Note: We subtract 3m on each side (left and right, or top and bottom), so we subtract 6 from the total length.
Outer area = a² Inner area = (a − 6)² Area of path = Outer area − Inner area = a² − (a−6)² = a² − (a² − 12a + 36) = 12a − 36 = 12(a − 3)
This simplified formula works for any square plot with a 3m wide path.
Given: Square plot of side 30m; path width = 3m
Step 1: Outer area = 30 × 30 = 900 m² Step 2: Inner side = 30 − 2×3 = 30 − 6 = 24m Step 3: Inner area = 24 × 24 = 576 m² Step 4: Area of path = 900 − 576 = 324 m²
Using formula: 12(30 − 3) = 12 × 27 = 324 m² ✓
Given: Square plot of side 20m; path width = 3m
Step 1: Outer area = 20 × 20 = 400 m² Step 2: Inner side = 20 − 6 = 14m Step 3: Inner area = 14 × 14 = 196 m² Step 4: Area of path = 400 − 196 = 204 m²
Using formula: 12(20 − 3) = 12 × 17 = 204 m² ✓
Inner side = 30 − 6 = 24m. Outer area = 900 m². Inner area = 576 m². Area of path = 900 − 576 = 324 m².
Area of path = Outer side² − (Outer side − 2w)² where w is the path width. For 3m path: Area = a² − (a−6)² = 12(a−3).
Inner side = 50 − 6 = 44m. Outer area = 2500 m². Inner area = 44² = 1936 m². Path area = 2500 − 1936 = 564 m². Or using formula: 12(50−3) = 12 × 47 = 564 m².
Because the 3m path runs along all four sides. On each axis (length and width), path occupies 3m from each end, so we subtract 3 + 3 = 6m total from the outer side.
Area of path = a² − (a−2w)² = 4w(a−w). For w=3: 4×3×(a−3) = 12(a−3).
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